60 citations · 154 across the 30 of their papers we have counts for
38 papers
Zonoids whose polars are zonoids: the Banach--Mazur distance need not tend to one
Dmitry Ryabogin, Artem Zvavitch
For every , we consider a Gaussian zonoid of revolution arising from works of Vitale and Mathis. We prove that is also a zonoid and compute the Banach--Mazur…
Simultaneous Busemann-Petty and Shephard Volume Comparisons
Artem Zvavitch
We study the simultaneous Busemann-Petty and Shephard volume comparison problem: whether comparison of the volumes of all central hyperplane sections and all orthogonal hyperplane…
Isomorphic Busemann--Petty for arbitrary measures: the sharp order
Alexander Koldobsky, Artem Zvavitch
Let be the optimal constant with the following property. For every even, continuous, strictly positive density on and all origin-symmetric convex bodies $K,L\subset…
Weighted Brunn-Minkowski Theory II: Inequalities for Mixed Measures and Applications
Matthieu Fradelizi, Dylan Langharst, Mokshay Madiman +1
In "Weighted Brunn-Minkowski Theory I", the prequel to this work, we discussed how recent developments on concavity of measures have laid the foundations of a nascent weighted Brun…
Comparison Problems for Radon Transforms
Alexander Koldobsky, Michael Roysdon, Artem Zvavitch
Given two non-negative functions and such that the Radon transform of is pointwise smaller than the Radon transform of , does it follow that the -norm of is…
Inequalities for sections and projections of convex bodies
Apostolos Giannopoulos, Alexander Koldobsky, Artem Zvavitch
This article belongs to the area of geometric tomography, which is the study of geometric properties of solids based on data about their sections and projections. We describe a new…
Volume Product
Matthieu Fradelizi, Mathieu Meyer, Artem Zvavitch
Our purpose here is to give an overview of known results and open questions concerning the volume product of a convex…
Weighted Brunn-Minkowski Theory I: On Weighted Surface Area Measures
Matthieu Fradelizi, Dylan Langharst, Mokshay Madiman +1
The Brunn-Minkowski theory in convex geometry concerns, among other things, the volumes, mixed volumes, and surface area measures of convex bodies. We study generalizations of thes…
On the volume of the Minkowski sum of zonoids
Matthieu Fradelizi, Mokshay Madiman, Mathieu Meyer +1
We explore some inequalities in convex geometry restricted to the class of zonoids. We show the equivalence, in the class of zonoids, between a local Alexandrov-Fenchel inequality,…
Sumset estimates in convex geometry
Matthieu Fradelizi, Mokshay Madiman, Artem Zvavitch
Sumset estimates, which provide bounds on the cardinality of sumsets of finite sets in a group, form an essential part of the toolkit of additive combinatorics. In recent years, pr…
General Measure Extensions of Projection Bodies
Dylan Langharst, Michael Roysdon, Artem Zvavitch
The inequalities of Petty and Zhang are affine isoperimetric-type inequalities providing sharp bounds for where is a projecti…
Inequalities for the Radon transform on convex sets
Apostolos Giannopoulos, Alexander Koldobsky, Artem Zvavitch
Several years ago the authors started looking at some problems of convex geometry from a more general point of view, replacing volume by an arbitrary measure. This approach led to…
Measure comparison and distance inequalities for convex bodies
Alexander Koldobsky, Grigoris Paouris, Artem Zvavitch
We prove new versions of the isomorphic Busemann-Petty problem for two different measures and show how these results can be used to recover slicing and distance inequalities. We al…
Brunn-Minkowski type inequalities for the lattice point enumerator
David Iglesias, Jesús Yepes Nicolás, Artem Zvavitch
Geometric and functional Brunn-Minkowski type inequalities for the lattice point enumerator are provided. In particular, we show that $$\mathrm{G}_n((1-λ)K +…
Geometry and volume product of finite dimensional Lipschitz-free spaces
Matthew Alexander, Matthieu Fradelizi, Luis C. García-Lirola +1
The goal of this paper is to study geometric and extremal properties of the convex body , which is the unit ball of the Lipschitz-free Banach space associated wi…
Volume of the Minkowski sums of star-shaped sets
Matthieu Fradelizi, Zsolt Lángi, Artem Zvavitch
For a compact set and an integer , let us denote by the Minkowski…
Equipartitions and Mahler volumes of symmetric convex bodies
Matthieu Fradelizi, Alfredo Hubard, Mathieu Meyer +2
Following ideas of Iriyeh and Shibata we give a short proof of the three-dimensional Mahler conjecture {\mf for symmetric convex bodies}. Our contributions include, in particular,…
On Rogers-Shephard type inequalities for general measures
David Alonso-Gutiérrez, María A. Hernández Cifre, Michael Roysdon +2
In this paper we prove a series of Rogers-Shephard type inequalities for convex bodies when dealing with measures on the Euclidean space with either radially decreasing densities,…
Wulff shapes and a characterization of simplices via a Bezout type inequality
Christos Saroglou, Ivan Soprunov, Artem Zvavitch
Inspired by a fundamental theorem of Bernstein, Kushnirenko, and Khovanskii we study the following Bezout type inequality for mixed volumes $$ V(L_1,\dots,L_{n})V_n(K)\leq V(L_1,K[…
Polytopes of Maximal Volume Product
Matthew Alexander, Matthieu Fradelizi, Artem Zvavitch
For a convex body , let $K^z = \{y\in{\mathbb R}^n : \langle y-z, x-z\rangle\le 1, \mbox{\ for all\ } x\in K\}$ be the polar body of with respect to th…
The convexification effect of Minkowski summation
Matthieu Fradelizi, Mokshay Madiman, Arnaud Marsiglietti +1
Let us define for a compact set the sequence $$ A(k) = \left\{\frac{a_1+\cdots +a_k}{k}: a_1, \ldots, a_k\in A\right\}=\frac{1}{k}\Big(\underset{k\ {\rm ti…
Estimating volume and surface area of a convex body via its projections or sections
Alexander Koldobsky, Christos Saroglou, Artem Zvavitch
The main goal of this paper is to present a series of inequalities connecting the surface area measure of a convex body and surface area measure of its projections and sections. We…
Distribution functions of sections and projections of convex bodies
Jaegil Kim, Vladyslav Yaskin, Artem Zvavitch
Typically, when we are given the section (or projection) function of a convex body, it means that in each direction we know the size of the central section (or projection) perpendi…
Characterization of Simplices via the Bezout Inequality for Mixed volumes
Christos Saroglou, Ivan Soprunov, Artem Zvavitch
We consider the following Bezout inequality for mixed volumes: It was shown…